On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence

In this paper, we consider in ${\mathbb{R}}^3$ the following zero mass Schrödinger-Bopp-Podolsky system $$\{\begin{array}{l}-\mathrm{\Delta }u+q^2\phi u=\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u\\ -\mathrm{\Delta }\phi +a^2{\mathrm{\Delta }}^2\phi =4\pi u^2\end{array}$$ where $a>0$, $q\mathrm{\ne }0$ and $p\in (3,6)$. Inspired by [Ruiz, Arch. Ration. Mech. Anal. 198 (2010)], we introduce a Sobolev space $\mathcal{E}$ endowed with a norm containing a nonlocal term. Firstly, we provide some fundamental properties for the space $\mathcal{E}$ including embeddings into Lebesgue spaces. Moreover a general lower bound for the Bopp-Podolsky energy is obtained. Based on these facts, by applying a perturbation argument, we finally prove the existence of a weak solution to the above system.